Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$6[x-(2-3 x)+1]=4 x-6$$

Step-by-Step Solution

Answer

The solution is \( x = 0 \).

Step 1: Simplify the Equation

First, we distribute the terms inside the brackets on the left side of the equation. The equation is: \[ 6[x - (2 - 3x) + 1] = 4x - 6 \]We first simplify inside the brackets:\[ x - (2 - 3x) + 1 = x - 2 + 3x + 1 = 4x - 1 \]Now, the equation becomes:\[ 6(4x - 1) = 4x - 6 \]

Step 2: Distribute and Rearrange

Next, distribute the 6 on the left side:\[ 6 imes 4x - 6 imes 1 = 24x - 6 \]Thus, the equation becomes:\[ 24x - 6 = 4x - 6 \]Subtract 4x from both sides to get:\[ 24x - 4x = 6 - 6 \]This simplifies to:\[ 20x = 0 \]

Step 3: Solve for x

To solve for \( x \), divide both sides of the equation by 20:\[ x = \frac{0}{20} = 0 \]Thus, the solution is \( x = 0 \).

Step 4: Check the Solution

Substitute \( x = 0 \) back into the original equation to check:First, substitute into left-hand side: \[ 6[0 - (2 - 3 imes 0) + 1] = 6[-1] = -6 \]Then substitute into right-hand side:\[ 4 imes 0 - 6 = -6 \]Since both sides equal \(-6\), \( x = 0 \) is indeed the correct solution.

Step 5: Graphical Verification

To verify graphically, plot both sides of the equation. The left side of the equation simplified is:\[ y = 6(4x - 1) = 24x - 6 \]And the right side is:\[ y = 4x - 6 \]Graph both equations. The intersection point is \( x = 0 \), confirming our solution.

Key Concepts

Equation Simplification

Equation simplification is the process of making an equation easier to understand or solve. It's like cleaning up your math room to find things more easily. In the given problem, we start by simplifying inside the brackets.

• Identify expressions that can be simplified, such as arithmetic operations or terms to be combined.
• In our equation, we have: \( x - (2 - 3x) + 1 \)
• Simplifying gives: \( x - 2 + 3x + 1 \)
• Combine like terms, turning it into: \( 4x - 1 \)

Breaking down the equation step-by-step helps avoid mistakes and makes the equation easier to work with. Once simplified, it's easier to see what operations are needed next.

Graphical Verification

Graphical verification involves using a graph to confirm the solution of an equation. It's like getting a visual "thumbs up" that you've done things right. Imagine plotting two lines on a graph corresponding to each side of your equation:

• For the left side of our simplified equation: \( y = 24x - 6 \)
• For the right side: \( y = 4x - 6 \)

By plotting these, you can visually see where they intersect.
The intersection point of the two lines gives the solution to the equation.
If both lines meet at \( x = 0 \), your solution is confirmed. Graphical verification is a powerful tool because it provides a clear and intuitive check for correctness.

Distributive Property

The distributive property is like a math rule for fairness. It helps distribute a number equally across terms inside parentheses. In our equation, this rule allows us to multiply the number outside the brackets by each term inside:

• For \( 6(4x - 1) \), distribute to get: \( 6 \times 4x - 6 \times 1 \)
• This simplifies to: \( 24x - 6 \)

Using the distributive property can greatly simplify equations and is crucial for proper equation handling.
Remember, it ensures that every component gets multiplied, which helps in preserving the equality of both sides.

Solve for Variable

Solving for a variable means isolating it on one side of the equation to find its value. It's like unwrapping a present to see what's inside! Using our previously simplified equation \( 24x - 6 = 4x - 6 \):

• Subtract \( 4x \) from both sides: \( 24x - 4x = 0 \)
• This simplifies further to: \( 20x = 0 \)
• Divide both sides by 20 to isolate \( x \): \( x = \frac{0}{20} = 0 \)

Remember, operations such as addition, subtraction, multiplication, or division should be equally applied to both sides.
This ensures that the equation remains balanced, leading you to the correct solution.